SOLUTION: If 10 samples (items) are chosen from a population with defect rate = 0.3, what is the probability that exactly 6 items are defective

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Question 1092301: If 10 samples (items) are chosen from a population with defect rate = 0.3, what is the probability that exactly 6 items are defective
Answer by jim_thompson5910(35256) (Show Source):
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n = 10 is the sample size
p = 0.3 is the defect rate
k = 6 number of items

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Plug n = 10 and k = 6 into the combination formula to get
n C k = (n!)/(k!*(n-k)!)
10 C 6 = (10!)/(6!*(10-6)!)
10 C 6 = (10!)/(6!*4!)
10 C 6 = (10*9*8*7*6!)/(6!*4!)
10 C 6 = (10*9*8*7)/(4!)
10 C 6 = (10*9*8*7)/(4*3*2*1)
10 C 6 = 5040/24
10 C 6 = 210

Now use that combination value to compute the binomial probability
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 6) = (10 C 6)*(0.3)^(6)*(1-0.3)^(10-6)
P(X = 6) = (10 C 6)*(0.3)^(6)*(0.7)^(4)
P(X = 6) = (210)*(0.3)^(6)*(0.7)^4
P(X = 6) = (210)*(0.000729)*(0.2401)
P(X = 6) = 0.036756909

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The final answer is 0.036756909 (this is equivalent to 3.6756909%)